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    <title>fstair</title>
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    <center>Scilab Function</center>
    <div align="right">Last update : April 1993</div>
    <p>
      <b>fstair</b> -  computes  pencil  column echelon form by qz transformations</p>
    <h3>
      <font color="blue">Calling Sequence</font>
    </h3>
    <dl>
      <dd>
        <tt>[AE,EE,QE,ZE,blcks,muk,nuk,muk0,nuk0,mnei]=fstair(A,E,Q,Z,stair,rk,tol)  </tt>
      </dd>
    </dl>
    <h3>
      <font color="blue">Parameters</font>
    </h3>
    <ul>
      <li>
        <tt>
          <b>A</b>
        </tt>: m x n matrix with real  entries.</li>
      <li>
        <tt>
          <b>tol</b>
        </tt>: real positive scalar.</li>
      <li>
        <tt>
          <b>E</b>
        </tt>: column echelon form matrix</li>
      <li>
        <tt>
          <b>Q</b>
        </tt>: m x m unitary matrix</li>
      <li>
        <tt>
          <b>Z</b>
        </tt>: n x n unitary matrix</li>
      <li>
        <tt>
          <b>stair</b>
        </tt>: vector of indexes (see ereduc)</li>
      <li>
        <tt>
          <b>rk</b>
        </tt>: integer, estimated rank of the matrix</li>
      <li>
        <tt>
          <b>AE</b>
        </tt>: m x n matrix with real  entries.</li>
      <li>
        <tt>
          <b>EE</b>
        </tt>: column echelon form matrix</li>
      <li>
        <tt>
          <b>QE</b>
        </tt>: m x m unitary matrix</li>
      <li>
        <tt>
          <b>ZE</b>
        </tt>: n x n unitary matrix</li>
      <li>
        <tt>
          <b>nblcks</b>
        </tt>:is the number of submatrices having full row rank &gt;= 0  detected in matrix  <tt>
          <b>A</b>
        </tt>.</li>
      <li>
        <tt>
          <b>muk:  </b>
        </tt>integer array of dimension (n). Contains the column dimensions mu(k)  (k=1,...,nblcks) of the submatrices having full column  rank in the pencil sE(eps)-A(eps)</li>
      <li>
        <tt>
          <b>nuk:  </b>
        </tt>integer array of dimension (m+1). Contains the row dimensions nu(k)  (k=1,...,nblcks) of the submatrices having full row  rank in the pencil sE(eps)-A(eps)</li>
      <li>
        <tt>
          <b>muk0:  </b>
        </tt>integer array of dimension (n). Contains the column dimensions mu(k)  (k=1,...,nblcks) of the submatrices having full column  rank in the pencil sE(eps,inf)-A(eps,inf)</li>
      <li>
        <tt>
          <b>nuk:  </b>
        </tt>integer array of dimension (m+1). Contains the row dimensions nu(k)  (k=1,...,nblcks) of the submatrices having full row  rank in the pencil sE(eps,inf)-A(eps,inf)</li>
      <li>
        <tt>
          <b>mnei:  </b>
        </tt>integer array of dimension (4). mnei(1) = row dimension of sE(eps)-A(eps)</li>
    </ul>
    <h3>
      <font color="blue">Description</font>
    </h3>
    <p>
    Given a pencil <tt>
        <b>sE-A</b>
      </tt> where matrix <tt>
        <b>E</b>
      </tt> is in column echelon form the
    function  <tt>
        <b>fstair</b>
      </tt> computes according to the wishes of the user a
    unitary transformed pencil <tt>
        <b>QE(sEE-AE)ZE</b>
      </tt> which is more or less similar
    to the generalized Schur form of the pencil <tt>
        <b>sE-A</b>
      </tt>.
    The function  yields also part of the Kronecker structure of
    the given pencil.</p>
    <p>
      <tt>
        <b>Q,Z</b>
      </tt> are the unitary matrices used to compute the pencil where E
    is in column echelon form (see ereduc)</p>
    <h3>
      <font color="blue">See Also</font>
    </h3>
    <p>
      <a href="quaskro.htm">
        <tt>
          <b>quaskro</b>
        </tt>
      </a>,&nbsp;&nbsp;<a href="ereduc.htm">
        <tt>
          <b>ereduc</b>
        </tt>
      </a>,&nbsp;&nbsp;</p>
    <h3>
      <font color="blue">Author</font>
    </h3>
    <p>Th.G.J. Beelen (Philips Glass Eindhoven). SLICOT</p>
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